Heights in the Isogeny Class of an Abelian Variety

Let A be an abelian variety over an algebraic closure of Q. A conjecture of Mocz asserts that there are only finitely many isomorphism classes of abelian varieties isogenous A, and of height less than some fixed constant c.

In this talk, I will sketch a proof of the conjecture when the Mumford-Tate conjecture - which is known in many cases - holds for A. This result should be compared with Faltings' famous theorem, which is about finiteness for abelian varieties defined over a fixed number field.

This is joint work with Lucia Mocz.

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